Set theory and its applications. L. Babinkostova, A. E. Caicedo, S. Geschke, M. Scheepers, eds. Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011. ISBN-10: 0-8218-4812-7 ISBN-13: 978-0-8218-4812-8

Here is a link to the AMS page for it, a link to its table of contents, and the preface:

The Boise Extravaganza in Set Theory (BEST) started in 1992 as a small, locally funded conference dedicated to Set Theory and its Applications. A number of years after its inception BEST started being funded by the National Science Foundation. Without this funding it would not have been possible to maintain the conference. The conference remained relatively small with many opportunities for its participants to meet informally. We like to think that during these years BEST has made it possible for the numerous set theorists who have participated in it to absorb, besides the new developments featured in the conference talks, also part of the folklore and traditions of the ﬁeld of set theory and its relatives. An explicit effort was made to bring together role models from various career stages in set theory as well as the new generation to support some notion of continuity in the ﬁeld.
This volume has a similar purpose. In it the reader will ﬁnd valuable papers ranging from surveys that put in print here set theoretic knowledge that has been around for several decades as unpublished lore, to hybrid survey-research papers, to pure research papers. Readers can be assured of the authority of each paper since each has been carefully refereed. The reader will also ﬁnd that the subjects treated in these papers range over several of the historically strongly represented areas of set theory and its relatives. Rather than expounding the virtues of each paper individually here, we invite the reader to learn from the authors.
Bringing to publication such a collection of papers is not possible without the generous dedication of authors and referees and the services of a publisher. We would like to thank all authors and referees for their selﬂess contributions to this volume. And we particularly would like to thank the publisher, Contemporary Mathematics, and Christine Thivierge, for the guidance they provided during this process.

43.614000-116.202000

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Nice sharing.
Many k-12 and undergraduate students think that Set theory is a basic thing made for children and has no application in advanced mathematics.
But in fact the set theory is the basis of the whole of mathematics.

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

The two concepts are different. For example, $\omega$, the first infinite ordinal, is the standard example of an inductive set according to the first definition, but is not inductive in the second sense. In fact, no set can be inductive in both senses (any such putative set would contain all ordinals). In the context of set theory, the usual use of the term […]

I will show that for any positive integers $n,\ell,k$ there is an $M$ so large that for all positive integers $i$, if $i/M\le \ell$, then the difference $$ \left(\frac iM\right)^n-\left(\frac{i-1}M\right)^n $$ is less than $1/k$. Let's prove this first, and then argue that the result follows from it. Note that $$ (i+1)^n-i^n=\sum_{k=0}^{n-1}\binom nk i^ […]

I think it is cleaner to argue without induction. If $n$ is a positive integer and $n\ge 8$, then $7n$ is both less than $n^2$ and a multiple of $n$, so at most $n^2-n$ and therefore $7n+1$ is at most $n^2-n+1

Let PRA be the theory of Primitive recursive arithmetic. This is a subtheory of PA, and it suffices to prove the incompleteness theorem. It is perhaps not the easiest theory to work with, but the point is that a proof of incompleteness can be carried out in a significantly weaker system than the theories to which incompleteness actually applies. It is someti […]

Here is a silly thing; I am not sure it is an "advantage" (or, for that matter, a disadvantage), but it indicates a difference: Inside a model $M$ of $\mathsf{ZF}$ there may be "hidden'' models $N$ of $\mathsf{ZF}$. The situation I have in mind is something like the following, which uses the fact that $\mathsf{ZF}$ is not finitely ax […]

Nice sharing.

Many k-12 and undergraduate students think that Set theory is a basic thing made for children and has no application in advanced mathematics.

But in fact the set theory is the basis of the whole of mathematics.

Are the BEST conferences still going on?

Hi Aaron. I expect so. We skipped a year.